- FindStat

Identifier

Mp00020: Binary trees —to Tamari-corresponding Dyck path⟶ Dyck paths
St000920: Dyck paths ⟶ ℤ

Values

[.,.] => [1,0] => 1

[.,[.,.]] => [1,1,0,0] => 1

[[.,.],.] => [1,0,1,0] => 1

[.,[.,[.,.]]] => [1,1,1,0,0,0] => 2

[.,[[.,.],.]] => [1,1,0,1,0,0] => 1

[[.,.],[.,.]] => [1,0,1,1,0,0] => 1

[[.,[.,.]],.] => [1,1,0,0,1,0] => 1

[[[.,.],.],.] => [1,0,1,0,1,0] => 1

[.,[.,[.,[.,.]]]] => [1,1,1,1,0,0,0,0] => 2

[.,[.,[[.,.],.]]] => [1,1,1,0,1,0,0,0] => 2

[.,[[.,.],[.,.]]] => [1,1,0,1,1,0,0,0] => 2

[.,[[.,[.,.]],.]] => [1,1,1,0,0,1,0,0] => 2

[.,[[[.,.],.],.]] => [1,1,0,1,0,1,0,0] => 1

[[.,.],[.,[.,.]]] => [1,0,1,1,1,0,0,0] => 2

[[.,.],[[.,.],.]] => [1,0,1,1,0,1,0,0] => 1

[[.,[.,.]],[.,.]] => [1,1,0,0,1,1,0,0] => 1

[[[.,.],.],[.,.]] => [1,0,1,0,1,1,0,0] => 1

[[.,[.,[.,.]]],.] => [1,1,1,0,0,0,1,0] => 2

[[.,[[.,.],.]],.] => [1,1,0,1,0,0,1,0] => 1

[[[.,.],[.,.]],.] => [1,0,1,1,0,0,1,0] => 1

[[[.,[.,.]],.],.] => [1,1,0,0,1,0,1,0] => 1

[[[[.,.],.],.],.] => [1,0,1,0,1,0,1,0] => 1

[.,[.,[.,[.,[.,.]]]]] => [1,1,1,1,1,0,0,0,0,0] => 2

[.,[.,[.,[[.,.],.]]]] => [1,1,1,1,0,1,0,0,0,0] => 2

[.,[.,[[.,.],[.,.]]]] => [1,1,1,0,1,1,0,0,0,0] => 2

[.,[.,[[.,[.,.]],.]]] => [1,1,1,1,0,0,1,0,0,0] => 2

[.,[.,[[[.,.],.],.]]] => [1,1,1,0,1,0,1,0,0,0] => 2

[.,[[.,.],[.,[.,.]]]] => [1,1,0,1,1,1,0,0,0,0] => 2

[.,[[.,.],[[.,.],.]]] => [1,1,0,1,1,0,1,0,0,0] => 2

[.,[[.,[.,.]],[.,.]]] => [1,1,1,0,0,1,1,0,0,0] => 2

[.,[[[.,.],.],[.,.]]] => [1,1,0,1,0,1,1,0,0,0] => 2

[.,[[.,[.,[.,.]]],.]] => [1,1,1,1,0,0,0,1,0,0] => 2

[.,[[.,[[.,.],.]],.]] => [1,1,1,0,1,0,0,1,0,0] => 2

[.,[[[.,.],[.,.]],.]] => [1,1,0,1,1,0,0,1,0,0] => 2

[.,[[[.,[.,.]],.],.]] => [1,1,1,0,0,1,0,1,0,0] => 2

[.,[[[[.,.],.],.],.]] => [1,1,0,1,0,1,0,1,0,0] => 1

[[.,.],[.,[.,[.,.]]]] => [1,0,1,1,1,1,0,0,0,0] => 2

[[.,.],[.,[[.,.],.]]] => [1,0,1,1,1,0,1,0,0,0] => 2

[[.,.],[[.,.],[.,.]]] => [1,0,1,1,0,1,1,0,0,0] => 2

[[.,.],[[.,[.,.]],.]] => [1,0,1,1,1,0,0,1,0,0] => 2

[[.,.],[[[.,.],.],.]] => [1,0,1,1,0,1,0,1,0,0] => 1

[[.,[.,.]],[.,[.,.]]] => [1,1,0,0,1,1,1,0,0,0] => 2

[[.,[.,.]],[[.,.],.]] => [1,1,0,0,1,1,0,1,0,0] => 1

[[[.,.],.],[.,[.,.]]] => [1,0,1,0,1,1,1,0,0,0] => 2

[[[.,.],.],[[.,.],.]] => [1,0,1,0,1,1,0,1,0,0] => 1

[[.,[.,[.,.]]],[.,.]] => [1,1,1,0,0,0,1,1,0,0] => 2

[[.,[[.,.],.]],[.,.]] => [1,1,0,1,0,0,1,1,0,0] => 1

[[[.,.],[.,.]],[.,.]] => [1,0,1,1,0,0,1,1,0,0] => 1

[[[.,[.,.]],.],[.,.]] => [1,1,0,0,1,0,1,1,0,0] => 1

[[[[.,.],.],.],[.,.]] => [1,0,1,0,1,0,1,1,0,0] => 1

[[.,[.,[.,[.,.]]]],.] => [1,1,1,1,0,0,0,0,1,0] => 2

[[.,[.,[[.,.],.]]],.] => [1,1,1,0,1,0,0,0,1,0] => 2

[[.,[[.,.],[.,.]]],.] => [1,1,0,1,1,0,0,0,1,0] => 2

[[.,[[.,[.,.]],.]],.] => [1,1,1,0,0,1,0,0,1,0] => 2

[[.,[[[.,.],.],.]],.] => [1,1,0,1,0,1,0,0,1,0] => 1

[[[.,.],[.,[.,.]]],.] => [1,0,1,1,1,0,0,0,1,0] => 2

[[[.,.],[[.,.],.]],.] => [1,0,1,1,0,1,0,0,1,0] => 1

[[[.,[.,.]],[.,.]],.] => [1,1,0,0,1,1,0,0,1,0] => 1

[[[[.,.],.],[.,.]],.] => [1,0,1,0,1,1,0,0,1,0] => 1

[[[.,[.,[.,.]]],.],.] => [1,1,1,0,0,0,1,0,1,0] => 2

[[[.,[[.,.],.]],.],.] => [1,1,0,1,0,0,1,0,1,0] => 1

[[[[.,.],[.,.]],.],.] => [1,0,1,1,0,0,1,0,1,0] => 1

[[[[.,[.,.]],.],.],.] => [1,1,0,0,1,0,1,0,1,0] => 1

[[[[[.,.],.],.],.],.] => [1,0,1,0,1,0,1,0,1,0] => 1

[.,[.,[.,[.,[.,[.,.]]]]]] => [1,1,1,1,1,1,0,0,0,0,0,0] => 2

[.,[.,[.,[.,[[.,.],.]]]]] => [1,1,1,1,1,0,1,0,0,0,0,0] => 2

[.,[.,[.,[[.,.],[.,.]]]]] => [1,1,1,1,0,1,1,0,0,0,0,0] => 2

[.,[.,[.,[[.,[.,.]],.]]]] => [1,1,1,1,1,0,0,1,0,0,0,0] => 2

[.,[.,[.,[[[.,.],.],.]]]] => [1,1,1,1,0,1,0,1,0,0,0,0] => 2

[.,[.,[[.,.],[.,[.,.]]]]] => [1,1,1,0,1,1,1,0,0,0,0,0] => 2

[.,[.,[[.,.],[[.,.],.]]]] => [1,1,1,0,1,1,0,1,0,0,0,0] => 2

[.,[.,[[.,[.,.]],[.,.]]]] => [1,1,1,1,0,0,1,1,0,0,0,0] => 2

[.,[.,[[[.,.],.],[.,.]]]] => [1,1,1,0,1,0,1,1,0,0,0,0] => 2

[.,[.,[[.,[.,[.,.]]],.]]] => [1,1,1,1,1,0,0,0,1,0,0,0] => 2

[.,[.,[[.,[[.,.],.]],.]]] => [1,1,1,1,0,1,0,0,1,0,0,0] => 2

[.,[.,[[[.,.],[.,.]],.]]] => [1,1,1,0,1,1,0,0,1,0,0,0] => 2

[.,[.,[[[.,[.,.]],.],.]]] => [1,1,1,1,0,0,1,0,1,0,0,0] => 2

[.,[.,[[[[.,.],.],.],.]]] => [1,1,1,0,1,0,1,0,1,0,0,0] => 2

[.,[[.,.],[.,[.,[.,.]]]]] => [1,1,0,1,1,1,1,0,0,0,0,0] => 2

[.,[[.,.],[.,[[.,.],.]]]] => [1,1,0,1,1,1,0,1,0,0,0,0] => 2

[.,[[.,.],[[.,.],[.,.]]]] => [1,1,0,1,1,0,1,1,0,0,0,0] => 2

[.,[[.,.],[[.,[.,.]],.]]] => [1,1,0,1,1,1,0,0,1,0,0,0] => 2

[.,[[.,.],[[[.,.],.],.]]] => [1,1,0,1,1,0,1,0,1,0,0,0] => 2

[.,[[.,[.,.]],[.,[.,.]]]] => [1,1,1,0,0,1,1,1,0,0,0,0] => 2

[.,[[.,[.,.]],[[.,.],.]]] => [1,1,1,0,0,1,1,0,1,0,0,0] => 2

[.,[[[.,.],.],[.,[.,.]]]] => [1,1,0,1,0,1,1,1,0,0,0,0] => 2

[.,[[[.,.],.],[[.,.],.]]] => [1,1,0,1,0,1,1,0,1,0,0,0] => 2

[.,[[.,[.,[.,.]]],[.,.]]] => [1,1,1,1,0,0,0,1,1,0,0,0] => 2

[.,[[.,[[.,.],.]],[.,.]]] => [1,1,1,0,1,0,0,1,1,0,0,0] => 2

[.,[[[.,.],[.,.]],[.,.]]] => [1,1,0,1,1,0,0,1,1,0,0,0] => 2

[.,[[[.,[.,.]],.],[.,.]]] => [1,1,1,0,0,1,0,1,1,0,0,0] => 2

[.,[[[[.,.],.],.],[.,.]]] => [1,1,0,1,0,1,0,1,1,0,0,0] => 2

[.,[[.,[.,[.,[.,.]]]],.]] => [1,1,1,1,1,0,0,0,0,1,0,0] => 2

[.,[[.,[.,[[.,.],.]]],.]] => [1,1,1,1,0,1,0,0,0,1,0,0] => 2

[.,[[.,[[.,.],[.,.]]],.]] => [1,1,1,0,1,1,0,0,0,1,0,0] => 2

[.,[[.,[[.,[.,.]],.]],.]] => [1,1,1,1,0,0,1,0,0,1,0,0] => 2

[.,[[.,[[[.,.],.],.]],.]] => [1,1,1,0,1,0,1,0,0,1,0,0] => 2

[.,[[[.,.],[.,[.,.]]],.]] => [1,1,0,1,1,1,0,0,0,1,0,0] => 2

[.,[[[.,.],[[.,.],.]],.]] => [1,1,0,1,1,0,1,0,0,1,0,0] => 2

[.,[[[.,[.,.]],[.,.]],.]] => [1,1,1,0,0,1,1,0,0,1,0,0] => 2

[.,[[[[.,.],.],[.,.]],.]] => [1,1,0,1,0,1,1,0,0,1,0,0] => 2

>>> Load all 1682 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

$F_{1} = q$

$F_{2} = 2\ q$

$F_{3} = 4\ q + q^{2}$

$F_{4} = 8\ q + 6\ q^{2}$

$F_{5} = 16\ q + 26\ q^{2}$

$F_{6} = 32\ q + 100\ q^{2}$

$F_{7} = 64\ q + 364\ q^{2} + q^{3}$

Description

The logarithmic height of a Dyck path.
This is the floor of the binary logarithm of the usual height increased by one:

$\lfloor\log_2(1+height(D))\rfloor$

Map

to Tamari-corresponding Dyck path

Description

Return the Dyck path associated with a binary tree in consistency with the Tamari order on Dyck words and binary trees.
The bijection is defined recursively as follows:

a leaf is associated with an empty Dyck path,
a tree with children $l,r$ is associated with the Dyck word $T(l) 1 T(r) 0$ where $T(l)$ and $T(r)$ are the images of this bijection to $l$ and $r$ .